Universal Multifractality Demonstrates Stable Topological Anderson Insulator Phase Boundaries

Scientists are increasingly focused on understanding how disorder impacts topological materials, and new research published this week details a significant step forward in that understanding. Ksenija Kovalenka, Ahmad Ranjbar, and Sam Azadi, alongside Rodion Vladimirovich Belosludov from the Institute for Materials Research at Tohoku University, et al., demonstrate the existence of a topological Anderson insulator, a phase stabilised by the interplay of disorder and topology, using the Haldane model. Their work maps the complete phase diagram of this insulator and, crucially, reveals universal critical spectra through multifractal analysis of low-energy eigenstates. This discovery unifies topology, localisation, and criticality, offering clear diagnostic tools for identifying disordered topological phases in real materials and potentially paving the way for robust topological devices.

This discovery unifies topology, localisation, and criticality, offering clear diagnostic tools for identifying disordered topological phases in real materials and potentially paving the way for robust topological devices.

Disorder Stabilises Topological Anderson Insulator Phase in disordered

This approach provides a robust method for identifying topological phases even in systems lacking translational symmetry, as the edge-integrated local Chern marker sensitively indicates topology. Experiments show that the critical spectra exhibit universal behaviour, remaining consistent regardless of whether disorder promotes or suppresses topological order. This finding is significant because it establishes a clear benchmark for real-space diagnostics of disordered topological phases, offering a powerful tool for characterizing these complex states of matter. The researchers achieved this by analysing how wave functions evolve at disorder-driven phase boundaries, employing multifractal analysis to characterise their statistical properties.

Complementing this, they performed finite-size multifractal analysis on eigenstates near the Fermi level, discovering that the generalized-dimension spectra collapse across a wide range of parameters at the TAI boundary. This collapse confirms a universal multifractal fingerprint of the critical states within this symmetry class, distinguishing it from the behaviour observed in clean topological or trivial insulating regimes. The Haldane model, defined by a tight-binding Hamiltonian on a honeycomb lattice, was used to explore these phenomena, incorporating nearest- and next-nearest-neighbor hopping, sublattice staggering, and on-site Anderson disorder. By tuning parameters such as mass and disorder strength, the researchers observed how the band structure and gap character evolved, providing insights into the transition between different insulating phases and the emergence of the TAI.

Mapping the Topological Anderson Insulator Phase Diagram

Researchers targeted the four eigenstates closest to E = 0 using the implicitly restarted Arnoldi method to analyse the multifractal properties of low-energy states at the phase boundaries. Exponents were extracted from ensemble averages of Pq over disorder realisations, treating each realisation equally. The team generated a representative map of D2 across the (M, W) plane, finding that the trivial BI region exhibited exponentially localised eigenstates with a vanishing correlation dimension of approximately 0. In contrast, the clean CI phase displayed wavefunctions strongly influenced by chiral boundary modes, resulting in a characteristic fractal dimension of around 1.5, indicative of edge contributions.

This edge-bulk imbalance provided a distinct multifractal fingerprint for the CI regime, differentiating it from the adjacent disorder-induced TAI. A continuous critical manifold was revealed, tracing the disorder-driven phase boundary with nearly identical D2 values, irrespective of the transition approach. To further investigate the multifractal structure, scientists analysed the q-dependence of scaling exponents τ(q) and generalised fractal dimensions Dq, averaging over 2000 disorder realisations at representative points in the (M, W) plane. The research showed that τ(q) exhibited nonlinearity in q, demonstrating genuine multifractal correlations in the wavefunctions.

The team fitted the numerical τ(q) spectra to a parabolic approximation, τ(q) = d(q −1) + γ q(1 −q), to extract anomalous multifractal exponents γ, summarised in Table I. In the CI phase, γ was approximately 0.321, consistent with edge-dominated critical states, while in the TAI regime, γ ranged from 0.500 to 0.650, reflecting the onset of disorder-induced localisation. Most remarkably, at the phase boundary, γ converged to 0.277, closely resembling the value of 0.262 reported for the integer quantum Hall plateau transition. This suggests the disorder-driven topological transition in the Haldane model belongs to the same unitary universality class as the quantum Hall critical point, linking lattice Chern insulators to the multifractal criticality of the IQHE.

Disorder Stabilises Topological Anderson Insulator Phase in disordered

Data shows that the multifractal spectra collapse across various points in the mass-disorder phase diagram along the TAI boundary, revealing a consistent multifractal fingerprint of the critical states. In contrast, the spectra within the clean topological regime exhibited edge dominance, with Dq approaching 1 for large positive q, while the trivial insulating regime displayed bulk localization and Dq tending towards 0. The study formulated the tight-binding Haldane model for spinless fermions on a honeycomb lattice, defined by a Hamiltonian incorporating nearest and next-nearest-neighbor hopping, sublattice staggering, and on-site Anderson disorder. All numerical values are expressed in units of t1, with disorder strength W ranging from -W/2 to W/2.

Analysis of nanoribbon band structures with open boundaries demonstrated how the gap character evolves with varying parameters, specifically M and t2. The team observed that the relative strength of the mass M and t2 dictates whether the clean system is topologically non-trivial or a conventional insulator. Tests prove that the edge-integrated local Chern marker effectively resolves the trivial, topological, and TAI regimes, even without translational symmetry. This breakthrough delivers a deeper understanding of the interplay between disorder and topology in condensed matter systems.

Disorder Stabilises a Novel Topological Anderson Insulator phase

This TAI represents a distinct phase made possible by the interplay of disorder and topology. The authors acknowledge that their study focuses on a specific model and that the observed universality may not extend to all disordered topological systems. Future research could explore the implications of these findings in more complex materials and geometries. Nevertheless, this work highlights general mechanisms by which disorder can both stabilize and destabilize topological phases, suggesting testable predictions for transport and spectroscopic measurements in engineered lattices and correlated materials. The disordered Haldane model, therefore, provides a valuable platform for understanding the intricate relationships between topology, localization, and criticality in condensed matter physics.